Conrey–Farmer–Keating–Rubinstein–Snaith conjecture for moments of real Dirichlet $L$-functions

Let $S=\{s_1,\dots,s_k\}$ be a set of complex numbers satisfying $|\operatorname{Re}(s_j)-1/2|\ll1/\log q$ and $|\operatorname{Im}(s_j)|\ll T^{1-\varepsilon}$. Let $f$ be a smooth function with compact support in $(0,\infty)$, and let the starred sum run over positive odd square-free integers $d$. Define $S_J=\{s_j:j\in J\}$, $S_J^{-}=\{1-s_j:j\in J\}$, and let $\widetilde f$ denote the Mellin transform of $f$. Set

$$T(S):=\frac{2}{3\zeta(2)}\sum_{\substack{n_1\cdots n_k=\square\\ n_1\cdots n_k\ \operatorname{odd}}}\frac{a(n_1\cdots n_k)}{n_1^{s_1}\cdots n_k^{s_k}},\qquad X(s):=\left(\frac{\pi}{8}\right)^{s-1/2}\frac{\Gamma\left(\frac{1-s}{2}\right)}{\Gamma\left(\frac{s}{2}\right)},$$

where

$$a(n)=\prod_{p\mid n}\left(1+\frac{1}{p}\right)^{-1}.$$

Conrey–Farmer–Keating–Rubinstein–Snaith conjecture. For some $\delta>0$,

$$\begin{aligned} \sum_{d\geq1}^{*}f\left(\frac{d}{X}\right)L(s_1,\chi_{8d})\cdots L(s_k,\chi_{8d}) &=\sum_{J\subset\{1,\dots,k\}}X^{1+\frac{|J|}{2}-\sum_{j\in J}s_j}\widetilde f\left(1+\frac{|J|}{2}-\sum_{j\in J}s_j\right)\\ &\quad\times\prod_{j\in J}X(s_j)\,T\left(S\setminus S_J\cup S_J^{-}\right)+O(X^{1-\delta}). \end{aligned}$$

Here the starred sum is over positive odd square-free integers, and the conjecture predicts a uniform asymptotic formula for shifted moments of real quadratic Dirichlet $L$-functions. The central-point moment conjecture of Keating and Snaith motivated this more precise number-theoretic formulation, which is based on the recipe of Conrey, Farmer, Keating, Rubinstein and Snaith. The supplied source does not state whether this formula has been proved or disproved.